Transformations
Apart from the already discussed time-frequency transformations, for systems theory the most important is the Möbius transformation and its use in applications relating to the Hardy spaces.
In geometry the Möbius transformation means the transformation of signal curve spaces plotted in the usual Cartesian (i.e. orthogonal) coordinate system by means of rotation, stretching and shrinking. This method helps us to transform signals running to infinity into a circle where our calculations can be further supported. This circle is the rotator of the inversion, such as feedback.
In the case of analytical functions the Möbius transformation keeps the properties which are relevant to the processes described by them. This is a group property from the aspect of the Galois group theory and is a critical condition for the analysis of the processes represented this way.
Calculability is supported by two previously highlighted consequences which are valid to analytical functions: one is that the roots of the characteristic equation, the polynomial equation that is similar to and belongs to the same groups as the differential equation describing the change of the process are identical with the roots of the differential equation, and thus we can calculate the poles determining the properties of the process, such as stability and other sensitivities. Poles are the unique singular working points of equations and thus of processes, so they should be avoided. In the knowledge of the poles, the describing differential equation can be represented as the product of multiplication, leading to the Blaschke product which provides the quality norms which will be important later for the control system. The absolute value of the difference elements of the Blaschke product also shows the distances from the critical poles.
This brings us back to the basic model: the system model which is the connection of the signal spaces of inputs and outputs by an operator system. So, the operator is a means to represent the system itself, its transformational behaviour, the set on which the operations are performed. In the basic model we model the signal transforming operation of the operator considering shift in time. This so-called shift operator conceptually corresponds to one of the basic calculation tools of dynamics, the infinitesimal calculation, which performs the change operation in small steps and sums these steps in an integrated way to assess the result of changes, the states of the system. This is how state representation connects to the world of signal spaces and operator spaces, to the Hankel and Toeplitz matrix operations. The successive matrix elements of changes represent hidden Markov processes in the stochastic approach, and the relations of the transformation matrices of the state space, i.e. the A, B, C, D representations in the state space approach.
For the space of operations we are looking for spaces that meet our requirements. Such space is the space of analytical functions. It is a requirement that it should have a calculable finite value both on the complex half-plane that meets the stability criterion (that is in the space which is insignificantly changing in time and is determined by the complex exponential function with negative exponent) and in its circle representation. These spaces are Hardy spaces, the spaces of finite analytical functions that can reach a maximum value and can be represented in the Euclidean and Lebesgue spaces. These properties and to a certain extent even further opportunities were discovered by Frigyes Riesz nearly a century ago, when the problems of today’s systems and control theory were hardly topical.
When representing things in a circle, the main aim is to achieve an analytical character and the possibility to bound with measures, but there is also a need for beauty of thought which comes together with lucidity. Here the circumference of the circle marks the suprema, the set of upper limits of processes represented within the circle as a boundary.
Of the two types of representation, the straight trajectory serves the course in time, while the circumference of the circle, including the adequate number of spinning around, serves the frequency interpretation. Remember that both representations show the processes of the real world only virtually but in a way that is suitable and practical for the calculations considering the given circumstances and tools.
This leads to the representation of processes in the Hardy spaces and within them to the supremal norm of Hinf which tends to infinity, that is, in practice it approximates a position of steady state, to the supreme norm. This norm integral in a way summarises the control tasks of the process, the domains of stability and the domains of robust operation of the process, and the potential quality properties of the process. The H2 and Hinf integral expression can be the starting formula of the z optimisation.
The visual, geometric implementation of the transformation of the two projective representations (on the complex plane and in the circle) are supported with the Möbius transformation which stretches, shrinks and rotates the shapes of the Euclidean space, as we can see on the example of a triangle in the visualisation of the Möbius transformation. The stability and alteration of group properties assist our thinking and calculations.
The relation between the two visual representations—the time function on the half-plane or the frequency-dependent representation in the circle—, which is isomorphic, isometric and invariant in terms of distances, is ensured by the Fourier-Laplace transformations, while the identity of the results (norms) is shown by the Parseval theorem.
It is worth giving a thought to another subtlety here, namely the difference between the Fourier and the Laplace transformations. The Laplace transformation transforms the Fourier formula by introducing the complex differential variable denoted by s and the shift value which modifies the changes and the energy relations of motions. The deep physical meaning of this shift was analysed by Emily Noether, one of the most brilliant mathematicians of the 20th century.
The figure of the flow of mathematical details shows the relations and transformations of our calculation spaces, the briefly presented Hardy spaces and the normed Lebesgue spaces (marked with l), which generalises distances with a function relation, and the relationships between the input and the output signal spaces, and the operating system status spaces.
As we have referred to it before, Hardy spaces use the quadratic and the above mentioned infinite degree norms: the former measure the quality properties, the latter measure the general and decisive properties of the process.
Now that we know the measure properties of the operator spaces, we can go back to the calculation tasks related to the function representations of the operator model, repeating the comments made on the analytical system of conditions from another point of view. Function representations also point at the fundamental correspondence between the content of classical and modern systems theory.
As we have emphasized before, the transformation functions working in the operator model are analytical ones and are infinitely differentiable, that is, they can be fully described with differential formulas. Linearity makes it possible to use the appropriate characteristic polynomial functional equations in the solution of these differential equations; the roots, the zero values are identical and can be calculated by identical algorithms and programmes. These roots—the zeros and the poles—are the critical irregular points of the system so they should be avoided when designing and setting up the system but they are important when dealing with other quality system properties, for example when manipulating the adaptation speed.
Thus, knowing the roots (poles and zeros) of system functions is a basic tool in control engineering: the function can be described as the fractional product of the roots where the absolute values of the product elements are the distances measured from the poles. The outstanding practical role of distance norms is again apparent here. The fractional product form takes us to the world of Blaschke products which provides further calculation possibilities.
As we have noted above, in the world of norms—and thus in technical practice—the second, original distance norm and the norms of infinity, the termination of the system operation have particular importance to us. The optimisation of these norms in the representations, and determining their position in the unit circle may be a control task.